We present a framework for carrying out global analyses of the Standard Model Effective Field Theory: SMEFiT. This approach is based on the Monte Carlo replica method, widely used in the case of NNPDF fits of the proton structure, for deriving a faithful estimate of the experimental and theoretical uncertainties. As a proof of concept of the SMEFiT methodology, we present a study of the constraints on the SMEFT provided by top quark production measurements from the LHC. We derive bounds for the 34 degrees of freedom relevant for the interpretation of the LHC top quark data and compare these bounds with previously reported constraints.